Piecewise smooth systems near a co-dimension 2 discontinuity manifold: Can one say what should happen?

Abstract

We consider a piecewise smooth system in the neighborhood of a co-dimension 2 discontinuity manifold Σ (intersection of two co-dimension 1 manifolds). Within the class of Filippov solutions, if Σ is attractive, one should expect solution trajectories to slide on Σ. It is well known, however, that the classical Filippov convexification methodology does not render a uniquely defined sliding vector field on Σ. The situation is further complicated by the possibility that, regardless of how sliding on Σ is taking place, during sliding motion a trajectory encounters so-called generic first order exit points, where Σ ceases to be attractive.In this work, we attempt to understand what behavior one should expect of a solution trajectory near Σ when Σ is attractive, what to expect when Σ ceases to be attractive (at least, at generic exit points), and finally we also contrast and compare the behavior of some regularizations proposed in the literature, whereby the original piecewise smooth system is replaced -in a neighborhood of Σ-by a smooth differential system.Through analysis and experiments in R 3 and R 4 , we will confirm some known facts, and provide some important insight: (i) when Σ is attractive, a solution trajectory indeed does remain near Σ, viz. sliding on Σ is an appropriate idealization (of course, in general, one cannot predict which sliding vector field should be selected); (ii) when Σ loses attractivity (at first order exit conditions), a typical solution trajectory leaves a neighborhood of Σ; (iii) there is no obvious way to regularize the system so that the regularized trajectory will remain near Σ as long as Σ is attractive, and so that it will be leaving (a neighborhood of) Σ when Σ looses attractivity.We reach the above conclusions by considering exclusively the given piecewise smooth system, without superimposing any assumption on what kind of dynamics near Σ (or sliding motion on Σ) should have been taking place. The only datum for us is the original piecewise smooth system, and the dynamics inherited by it.1991 Mathematics Subject Classification. 34A36, 65P99.

Limit cycles for regularized piecewise smooth systems with a switching manifold of codimension two

“…This lemma actually tells us the uniform convergence of the flow for regularization PWS to the flow of original PWS system when it has a codimension one switching manifold. However, this result is not sufficient to deal with our case, because our PWS system not only has codimension one switching manifolds Σ 1 and Σ 2 , but also has a codimension two switching manifold Σ. Fortunately, we have similar results for the codimension two case, the similar result has been proved in [8] for a given cubic transition function. We state this result as follows.…”

“…Lemma 18 in [11] and the results in [8] insures that P ǫ is well defined and it implies the following Proposition.…”

“…In [8], authors have proved that if Σ is attractive in finite time, the solution of regularized system (34) converges uniformly to the sliding solution with bilinear vector filed (15) on intervals of the form [0, T ]. Moreover, this result holds for general transition function.…”

Periodic orbits for double regularization of piecewise smooth systems with a switching manifold of codimension two

“…(5) was first seen in [1] and then studied by many authors, see e.g. [12]. Panazzolo and da Silva considered a multi-parameter regularization in [29], where they applied two monotonic transition functions.…”

Limit cycles for regularized discontinuous dynamical systems with a hyperplane of discontinuity

“…Finally, we stress that, our switching manifold is a codimension one plane. The case of higher co-dimension switching manifold (say, the intersection of two planes) is considerably more complex (e.g., see [7]), and remains to be considered as well.…”